7. Systems of Equations & Matrices

In Exercises 19–20, a few steps in the process of simplifying the given matrix to row-echelon form, with 1s down the diagonal from upper left to lower right, and 0s below the 1s, are shown. Fill in the missing numbers in the steps that are shown.

Write the system of equations associated with each augmented matrix . Do not solve.

Let and . Solve each matrix equation for X. 2X + A = B

In Exercises 14–27, perform the indicated matrix operations given that and D are defined as follows. If an operation is not defined, state the reason. -5(A+D)

Use the Gauss-Jordan method to solve each system of equations. For systems in two variables with infinitely many solutions, write the solution with y arbitrary. For systems in three variables with infinitely many solutions, write the solution set with z arbitrary.

x + y = 5

x - y = -1

Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

$\begin{cases}x + y - z = -2 \\2x - y + z = 5 \\-x + 2y + 2z = 1\end{cases}$

Use the Gauss-Jordan method to solve each system of equations. For systems in two variables with infinitely many solutions, write the solution with y arbitrary. For systems in three variables with infinitely many solutions, write the solution set with z arbitrary.

3x + 2y = -9

2x - 5y = -6

Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

$\begin{cases}x + 3y = 0 \\x + y + z = 1 \\3x - y - z = 11\end{cases}$

Perform the indicated matrix operations given that and D are defined as follows. If an operation is not defined, state the reason. BD

Solve each system, using the method indicated.

5x + 2y = -10

3x - 5y = -6 (Gauss-Jordan)

Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

$\begin{cases}2x - y - z = 4 \\x + y - 5z = -4 \\x - 2y = 4\end{cases}$

Use the Gauss-Jordan method to solve each system of equations. For systems in two variables with infinitely many solutions, write the solution with y arbitrary. For systems in three variables with infinitely many solutions, write the solution set with z arbitrary.

6x - 3y - 4 = 0

3x + 6y - 7= 0

Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

$\begin{cases}x + y + z = 4 \\x - y - z = 0 \\x - y + z = 2\end{cases}$

Use the Gauss-Jordan method to solve each system of equations. For systems in two variables with infinitely many solutions, write the solution with y arbitrary. For systems in three variables with infinitely many solutions, write the solution set with z arbitrary.

2x - y = 6

4x - 2y = 0

Solve each system, using the method indicated.

3x + y = -7

x - y = -5 (Gaussian elimination)